Question

a canadian coast guard rescue boat is searching for a ship in distress off the coast of newfoundland. a spotter notices a flare that has been fired vertically from the ship in distress. the height of the flare, h metres above the ocean, after t seconds is modelled by the quadratic function h(t)= - 4.9t² + 167.8t. answer the following accurate to one decimal place. a) how high is the flare after 4 seconds? use a graphing calculator for these questions h=-4.9(4)² + 167.8(4) h=-78.4 + 671.2 h = 592.8 m b) for how much time is the flare above 1 km? y1=-4.9x² + 167.8x y2 = 1000 (1000m = 1km) window x(0,40) y(0,2000) 1st intersection point=(7.683348,1000) 2nd intersection point=(26.56155,1000) >1 km = 26.56155 - 7.683348 = 18.878202 ≈ 18.9 s c) how much time does the flare take to reach its maximum height?

Explanation:

Step1: Calculate height at \(t = 4\)

Substitute \(t = 4\) into \(h=-4.9t^{2}+167.8t\).
\[h=-4.9(4)^{2}+167.8(4)\]
\[h=-4.9\times16 + 671.2\]
\[h=-78.4+671.2\]
\[h = 592.8\]

Step2: Solve for \(t\) when \(h = 1000\)

Set \(h=-4.9t^{2}+167.8t = 1000\), then \(4.9t^{2}-167.8t + 1000=0\).
Using the quadratic formula \(t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) where \(a = 4.9\), \(b=-167.8\), \(c = 1000\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-167.8)^{2}-4\times4.9\times1000\)
\[=28156.84 - 19600=8556.84\]
Then \(t=\frac{167.8\pm\sqrt{8556.84}}{9.8}=\frac{167.8\pm92.5}{9.8}\)
We get \(t_1=\frac{167.8 + 92.5}{9.8}=\frac{260.3}{9.8}\approx26.6\) and \(t_2=\frac{167.8-92.5}{9.8}=\frac{75.3}{9.8}\approx7.7\)

Step3: Find time to reach maximum height

For a quadratic function \(y = ax^{2}+bx + c\), the \(x\) - coordinate of the vertex (in our case \(t\) - coordinate for maximum of \(h=-4.9t^{2}+167.8t\)) is \(t=-\frac{b}{2a}\).
Here \(a=-4.9\), \(b = 167.8\), so \(t=-\frac{167.8}{2\times(-4.9)}=\frac{167.8}{9.8}\approx17.1\)

Answer:

a) The height of the flare after 4 seconds is \(592.8\) m.
b) The flare is above 1 km (\(1000\) m) at approximately \(t = 7.7\) s and \(t = 26.6\) s.
c) The flare takes approximately \(17.1\) s to reach its maximum height.